Keywords

MSC

37A3047A3503E72

1 Introduction

Ergodic theory is currently rapidly and massively developing area of theoretical and applied mathematical research. Ergodic theory theorems are studied in many structures, especially, in structures created on the basis of fuzzy approach. Some ergodic theorems valid in the classical ergodic theory [2] have been proven, among others, for D-posets of fuzzy sets [3], for MV-algebras of fuzzy sets [4, 5], and recently also for families of IF-events [6]. In our previous paper [1], we proved fuzzy versions of Mesiar’s ergodic theorems for the case of fuzzy dynamical systems defined and studied in [7]. This way, we contributed to the extension of our study of fuzzy dynamical systems. By a fuzzy dynamical system we mean a system \((\Omega, M, m, U)\), where \((\Omega, M, m)\) is a fuzzy probability space defined by Piasecki [8], and \(U: M \to M\) is an m-invariant σ-homomorphism. This structure can serve as an alternative mathematical model of ergodic theory for the case where the observed events are described vaguely. Fuzzy dynamical systems include the classical dynamical systems; on the other hand, they enable one to study more general situations. The aim of this paper is to generalize some other assertions valid in the classical ergodic theory to the case of fuzzy dynamical systems. In Section 3, after the introductory section (Section 2), we prove fuzzy analogies of Birkhoff’s individual ergodic theorem and the maximal ergodic theorem. The basic idea of our proofs is based on a factorization of the fuzzy σ-algebra M and on properties of the σ-homomorphism U. Note that other approaches to a fuzzy generalization of the notion of a dynamical system are presented in [9–12]. The authors of these papers used some other connectives to define the fuzzy set operations.

The symbols \(\bigvee_{n = 1}^{\infty} f_{n}: = \sup_{n}f_{n}\) and \(\bigwedge_{n = 1}^{\infty} f_{n} =:\inf_{n}f_{n}\) denote the fuzzy union and fuzzy intersection of a sequence \(\{ f_{n} \}_{n = 1}^{\infty} \subset M\), respectively, in the sense of Zadeh [13]. A couple \((\Omega, M)\), where Ω is a nonempty set, and M is a fuzzy σ-algebra of fuzzy subsets of Ω, is called a fuzzy measurable space. The presented σ-additive fuzzy measure m satisfies all properties analogous to the properties of classical probability measure in the crisp case. The described structure \((\Omega, M, m)\) can serve as a mathematical model of random experiments, the results of which are vaguely defined events, the so-called fuzzy events. A probability interpretation of the above notions is as follows: a set Ω is the set of elementary events; a fuzzy set from the system M is a fuzzy event; the value \(m(f)\) is the probability of a fuzzy event f; a fuzzy event \(f'\) is the opposite event to a fuzzy event f; and weakly separated fuzzy events are interpreted as mutually exclusive events.

is an m-invariant σ-homomorphism, and the quadruplet \((\Omega, M, m, U)\) is a fuzzy dynamical system.

Example 2.1

Let \((\Omega, S, P, T)\) be a classical dynamical system. Put \(M = \{ \chi_{A}; A \in S \}\), where \(\chi_{A}\) is the indicator of a set \(A \in S\), and define the mapping \(m:M \to \langle 0, 1 \rangle\) by \(m(\chi_{A}) = P(A)\). Then the triplet \((\Omega, M, m)\) is a fuzzy probability space, and the system \((\Omega, M, m, U)\), where the mapping \(U: M \to M\) is defined by (2.1), is a fuzzy dynamical system. By this procedure a classical model can be imbedded into a fuzzy one.

We can also consider the following extension of a fuzzy dynamical system from the previous example.

Example 2.2

Let \((\Omega, S, P, T)\) be a classical dynamical system. Denote by M the system of fuzzy subsets f of Ω such that f is an S-measurable mapping and \(P(f \in (1 / 4; 3 / 4)) = 0\). If we define the mapping \(m:M \to \langle 0, 1 \rangle\) by the equality \(m(f) = P\) (\(f > (1 / 2)_{\Omega} \)), then the triplet \((\Omega, M, m)\) is a fuzzy probability space, and the system \((\Omega, M, m, U)\), where the mapping \(U: M \to M\) is defined by (2.1), is a fuzzy dynamical system.

where \(B(\Re )\) is the family of all Borel subsets of the real line ℜ, and \(E^{C}\) denotes the complement of a set \(E \subset \Re\).

Example 2.4

Let \((\Omega, S, P)\) be a classical probability space, and \(\xi: \Omega \to \Re\) be a random variable in the sense of classical probability theory. Then the mapping x defined by \(x(E) = \chi_{\xi^{ - 1}(E)}\), \(E \in B(\Re )\), is an F-observable on the fuzzy measurable space \((\Omega, M)\) from Example 2.1.

Let x be an F-observable on a fuzzy measurable space \((\Omega, M)\). Then the range of F-observable x, that is, the set \(R(x): = \{ x(E); E \in B(\Re )\}\) is a Boolean σ-algebra of \((\Omega, M)\) with minimal and maximal elements \(x( \varnothing)\) and \(x(\Re )\), respectively. If \(U: M \to M\) is a σ-homomorphism, then it is easy to verify that the mapping \(U \circ x:B(\Re ) \to M\), \(U \circ x: E \to U (x (E))\), \(E \in B(\Re )\), is an F-observable on \((\Omega, M)\), too.

Let any fuzzy probability space \((\Omega, M, m)\) be given. If x is an F-observable on \((\Omega, M)\), then the mapping \(m_{x}: E \mapsto m(x(E))\), \(E \in B(\Re )\), is a probability measure on \(B(\Re )\). The value \(m(x(E))\) is interpreted as the probability that an F-observable x has a value in \(E \in B(\Re )\). By an integral of x with respect to m we mean the expression

$$m(x) = \int x \,d m: = \int_{\Re} t \,d m_{ x}(t) $$

(if the integral on the right-hand side exists and is finite). The integral \(m(x)\) is interpreted as the mean value of x [15]. Further, for a Borel-measurable function \(\psi: \Re \to \Re\), we put

Definition 2.4

Let \((\Omega, M, m)\) be a given fuzzy probability space, and \(x,x_{1},x_{2},\ldots\) be F-observables on \((\Omega, M)\). We say that the sequence \(\{ x_{n} \}_{n = 1}^{\infty}\) converges to x almost everywhere in m (and we write \(x_{n} \to x\) a.e. in m) if, for every \(\varepsilon > 0\),

3 Main results

In this section, we present Birkhoff’s individual ergodic theorem and the maximal ergodic theorem for fuzzy dynamical systems. In the proofs, we will use the factorization of a fuzzy σ-algebra M described further and the properties of a σ-homomorphism U. The presented results can be obtained also using the factorization over the σ-ideal of weakly empty sets. Details on this approach can be found in [17].

Then it is easy to see that h is a σ-homomorphism from M onto \([M]\) and \(\bar{x}: = h \circ x\) is an observable on the Boolean σ-algebra \([M]\).

In the following, we will need the notion of an ergodic fuzzy dynamical system. We introduce this notion analogously as in the classical ergodic theory [2].

Definition 3.1

A fuzzy dynamical system \((\Omega, M, m, U)\) is said to be ergodic if a σ-homomorphism U of \((\Omega, M)\) is ergodic in m, that is, for every \(f \in M\), the statement \(m(f \wedge U(f')) = 0 = m(U(f) \wedge f')\) implies \(m(f) \in \{ 0, 1 \}\).

The following theorem is a fuzzy analogue of Birkhoff’s individual ergodic theorem. It should be noted that the first authors interested in the ergodic theory on fuzzy measurable spaces were Harman and Riečan [18]. They proved Birkhoff’s individual ergodic theorem for the compatible case. Theorem 3.1 presents a more general case.

whereois the question observable of the empty fuzzy set\(0_{\Omega}\).

Proof

At the beginning of the proof, we use arguments similar to those in the proof of Theorem 3.1 in [1]. Let \(\mathcal{A}\) be the minimal Boolean sub-σ-algebra of \([M]\) containing all ranges of \(\bar{U}^{n} \circ \bar{x}\), \(n = 1, 2,\ldots\) . Then \(\bar{U}( [f]) \in \mathcal{A}\) for any \([f] \in \mathcal{A}\) and the Boolean σ-algebra \(\mathcal{A}\) has a countable generator. Therefore, in view of Varadarajan [19], there exists an observable \(z: B(\Re ) \to [M]\) such that \(\{ z (E): E \in B(\Re) \} = \mathcal{A}\). Moreover, there is a sequence of real-valued Borel functions \(\{ \psi_{n} \}_{n = 0}^{\infty}\) such that

4 Conclusions

In the classical theory, an event is understood as an exactly defined phenomenon, and from the mathematical point of view, it is a classical set. In practice, however, we often encounter events that are described imprecisely, vaguely, so-called fuzzy events. That is why various proposals for a fuzzy generalization of the dynamical system of classical ergodic theory have been created (e.g., in [7, 9–12]). In this note, we contributed to the extension of our study concerning fuzzy dynamical systems introduced by Markechová in [7]. We presented generalizations of Birkhoff’s individual ergodic theorem and the maximal ergodic theorem from the classical ergodic theory to the case of fuzzy dynamical systems.

Declarations

Acknowledgements

The authors thank the editor and the referees for their valuable comments and suggestions.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra